Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction
Abstract
We consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction. Subsequently, we study the validity of the so-called Smoluchowski-Kramers diffusion approximation. We show that, under more stringent conditions on the friction, in the small-mass limit the solution of the system of stochastic damped wave equations converges to the solution of a system of stochastic quasi-linear parabolic equations. In this convergence, an additional drift emerges as a result of the interaction between the noise and the state-dependent friction. The identification of this limit is achieved by using a suitable generalization of the classical method of perturbed test functions, tailored to the current infinite dimensional setting.
Keywords
Cite
@article{arxiv.2312.08925,
title = {Smoluchowski-Kramers diffusion approximation for systems of stochastic damped wave equations with non-constant friction},
author = {Sandra Cerrai and Arnaud Debussche},
journal= {arXiv preprint arXiv:2312.08925},
year = {2023}
}